Macroscopic model for solvated ion dynamics

Abstract

A macroscopic treatment of solvated ion dynamics is developed and applied to calculate the limiting (zero concentration) conductance of cations in several aprotic solvents. The theory is based on a coupled set of electrostatic and hydrodynamic equations for the density, flow, and polarization fields induced in the polar solvent by a moving ion. These equations, which are derived by the Mori projection technique, include crucial local solvent structure (ion solvation) effects through solvent compressibility, and local constitutive parameters. If solvent structure is suppressed, the equations reduce to those derived previously by Onsager and Hubbard [J. B. Hubbard and L. Onsager, J. Chem. Phys. 67, 4850 (1977)]. The macroscopic equations are approximately decoupled into electrostatic and hydrodynamic parts. The decoupled equations are solved assuming a step density, viscosity, and dielectric constant model for the local solvent structure and dynamics. This yields analytic expressions for the viscous, ζ_V, and dielectric ζ_D, contributions to the ion friction coefficient. These expressions generalize, respectively, the Stokes and Zwanzig results for the (slip) viscous and dielectric friction so as to account for ion solvation effects. The friction coefficients involve a desolvation function Δ which depends on the local structure (density) and dynamics of the solvent. The drag coefficient results reduce in form to those of Zwanzig (within a flow gradient correction factor of 2/3) and Stokes for both weak (Δ→1) and strong (Δ→0) ion–solvent interaction. For Δ→1 the true ionic radius R_i appears in the drag formulas while for Δ→0 a renormalized solvated ion radius σ=R_i+2R_s (where R_s=solvent molecule radius) appears. The theory is fit to experimental cation conductances in pyridine, acetone, and acetonitrile by representing Δ by a two parameter switching function. Agreement between the model and experiment is satisfactory for all three solvents. Moreover dielectric friction is found to have a negligible effect on ion mobility in these solvents.